In Waco and Surrounding Counties
Heart Disease has been the leading cause of death from 2009-20191
For perspective, during the height of COVID, it accounted for 1 in 8 deaths (or 697,000 deaths) in the United States. Heart disease was the number one cause of death, followed by cancer, with these two causes of death accounting for a total of 2.15 million deaths.2
Other research has shown other factors to impact heart disease fatalities:
Space
Time
Age
Gender
Ozone
Housing
Violent Crime
Mean estimated 8-hour average ozone concentration in parts per billion (ppb) within 3 meters of the surface of the earth
Fit Multivariate Spatial Models
– Linear Regression
– Spatial Model, no nugget effect {bbstdr}
– Spatial Model, nugget effect {spBayes} and {INLA}
Analyse Models
– Error Metrics
– Covariate Significance
– Acknowledge Computational Time
Fit Spatio-Temporal Models
– Spatio-temporal model fitting with {spTimer}
– Autoregressive model fitting with {INLA}
\(y|x ∼ N(\beta_0 + \beta_i(x), \sigma^2)\)
\(\beta_i \sim N(0,10000)\)
\(\sigma^2 \sim Gamma(\alpha = 2,\beta = 1)\)
| Params | mean | 2.5% | 97.5% | sd | Significant |
|---|---|---|---|---|---|
| (Intercept) | −74.7 | −25,912.3 | 25,763.0 | 13,173.9 | |
| 35_64 | −748.3 | −23,123.1 | 21,626.5 | 11,408.3 | |
| 65_and_older | 673.6 | −21,701.2 | 23,048.5 | 11,408.3 | |
| men | 108.8 | −22,266.0 | 22,483.6 | 11,408.3 | |
| women | −183.5 | −22,558.3 | 22,191.3 | 11,408.3 | |
| ozone | 1.1 | −12.3 | 14.4 | 6.8 | |
| hcp | 0.6 | −4.9 | 6.0 | 2.8 | |
| hpi | −0.7 | −1.0 | −0.4 | 0.2 | * |
| crime_rate | 2.1 | 1.6 | 2.6 | 0.2 | * |
| sigma2 | 34,704.3 | 29,651.5 | 40,603.5 | 2,796.6 |
{bmstdr}
\(Y \sim N_n(X\beta,\sigma^2_ϵH)\)
\(H_{ij} = exp(−\phi d_{ij})\)
\(d_{ij}\): \(s_i\) to \(s_j\)
\(\phi=0.9\)
| Params | mean | 2.5% | 97.5% | sd | Significant |
|---|---|---|---|---|---|
| (Intercept) | −73.0 | −26,027.4 | 25,881.5 | 13,233.4 | |
| 35_64 | −750.3 | −23,226.3 | 21,725.6 | 11,459.8 | |
| 65_and_older | 677.6 | −21,798.4 | 23,153.5 | 11,459.8 | |
| men | 110.8 | −22,365.2 | 22,586.7 | 11,459.8 | |
| women | −183.7 | −22,659.6 | 22,292.3 | 11,459.8 | |
| ozone | 1.3 | −11.9 | 14.4 | 6.7 | |
| hcp | 0.1 | −5.2 | 5.5 | 2.7 | |
| hpi | −0.7 | −1.0 | −0.4 | 0.2 | * |
| crime_rate | 2.1 | 1.6 | 2.6 | 0.2 | * |
| sigma2 | 35,018.8 | 29,920.2 | 40,971.4 | 2,821.9 |
{spBayes}
\(Y \sim N(X\beta, \sigma^2_{\epsilon}I+\tau^2_{\omega}S_{\omega})\)
\(S_{\omega} \sim exp(−\phi d_{ij})\)
\(\phi \sim Unif(a,b)\)
\(\sigma^2 \sim Gamma(\alpha_\sigma,\beta_\sigma)\)
\(\tau^2 \sim Gamma(\alpha_\sigma,\beta_\sigma)\)
| Params | mean | 2.5% | 97.5% | sd | Significant |
|---|---|---|---|---|---|
| X.Intercept. | −9.3 | −199.9 | 179.8 | 97.8 | |
| 35_64 | −700.8 | −838.3 | −561.2 | 71.2 | * |
| 65_and_older | 689.0 | 548.4 | 826.4 | 70.6 | * |
| men | 135.8 | −0.2 | 273.4 | 70.2 | |
| women | −149.9 | −287.9 | −15.8 | 70.1 | * |
| ozone | −1.3 | −8.2 | 5.8 | 3.6 | |
| hcp | 0.9 | −4.3 | 6.0 | 2.6 | |
| hpi | −0.7 | −1.0 | −0.4 | 0.2 | * |
| crime_rate | 2.0 | 1.6 | 2.4 | 0.2 | * |
| sigma.sq | 1.1 | 0.2 | 4.8 | 2.4 | |
| tau.sq | 35,617.3 | 30,295.5 | 41,940.1 | 3,079.3 | |
| phi | 0.9 | 0.0 | 1.9 | 0.6 |
{INLA}
\(Y \sim N(X\beta, \sigma^2_{\epsilon}I+\tau^2_{\omega}S_{\omega})\)
\(S_{\omega} \sim exp(−\phi d_{ij})\)
\(\phi \sim Unif(a,b)\)
\(\sigma^2 \sim Gamma(\alpha_\sigma,\beta_\sigma)\)
\(\tau^2 \sim Gamma(\alpha_\sigma,\beta_\sigma)\)
| Params | mean | 2.5% | 97.5% | sd | Significant |
|---|---|---|---|---|---|
| X.Intercept. | 0.4 | −61.6 | 62.9 | 31.6 | |
| 35_64 | −307.9 | −370.5 | −244.7 | 32.0 | * |
| 65_and_older | 306.4 | 244.4 | 368.9 | 31.6 | * |
| men | 63.4 | 8.2 | 118.3 | 27.9 | * |
| women | −63.3 | −119.0 | −8.3 | 28.5 | * |
| ozone | −1.7 | −13.3 | 9.6 | 5.9 | |
| hcp | 1.0 | −11.5 | 13.3 | 6.2 | |
| hpi | −0.7 | −1.4 | 0.0 | 0.4 | |
| crime_rate | 2.0 | 1.1 | 2.9 | 0.5 | * |
| phi | 1.0 | 0.4 | 2.4 | 0.6 | |
| sigmasq | 0.0 | 0.0 | 0.0 | 0.0 | |
| tausq | 209,301.1 | 168,267.5 | 248,651.5 | 20,436.8 |
| model | rmse | mae | cvg | Comp_Time | Sig |
|---|---|---|---|---|---|
| spBayes | 206.1 | 178.5 | 97.9 | 300 | men, ozone, housing_change_percent |
| none | 221.3 | 194.3 | 93.6 | 10 | 35_64, 65_and_older, men, women, ozone, housing_change_percent |
| no nugget | 238.6 | 204.5 | 89.3 | 17 | 35_64, 65_and_older, men, women, ozone |
| inla | 270.9 | 237.5 | 100.0 | 36 | ozone, housing_change_percent, hpi |
Move forward with {spBayes}
Location
Time
Age
Gender
Ozone
Housing Change Percent
\[Y_{nt} \sim N(X_{nt}\beta, \tau_{\omega}^2A_{nt}S_{\omega}A_{nt}'+ \sigma_{\epsilon}^2I)\] \(t = 1,...,T\)
\(Y_{nt}\): Observations
\(X_{nt}\): Covariate values
\(n_t\): locations at time \(t\)
\[A_t = C_tS^{-1}_{\omega}\] where \(S_{\omega}\) is \(m \times m\): Gaussian process
\(C_t\) is \(n_t \times m\) with the \(j\)th row and \(k\)th column entry \(exp(−\phi|s_j−s^*_k|)\) for \(j=1,...,n_t\) and \(k=1,...,m\).
\(C_t\) captures the cross-correlation between the observation locations at time \(t\) and the \(m\) locations, \(s^∗_k\) \(k=1,...,m\)
\(\beta \sim N(0,D), D = 1000\)
\(\sigma^2_{\epsilon} \sim Gamma(\alpha_\sigma,\beta_\sigma)\)
\(\tau^2_{\omega} \sim Gamma(\alpha_\tau,\beta_\tau)\)
\(\phi \sim Unif(0,T)\)
{spTimer}
\(\sigma^2_{\epsilon}\) random effect error
\(\sigma^2_{\eta}\) spatial error
\(\phi\) decay parameter
| Params | mean | 2.5% | 97.5% | sd | Significant |
|---|---|---|---|---|---|
| (Intercept) | 91.6 | −102.8 | 280.6 | 96.6 | |
| 35_64 | −638.1 | −770.3 | −493.2 | 69.8 | * |
| 65_and_older | 730.5 | 600.0 | 872.7 | 69.8 | * |
| men | 194.4 | 55.5 | 324.6 | 70.4 | * |
| women | −98.8 | −242.8 | 38.7 | 71.3 | |
| ozone | 15.0 | 8.8 | 21.1 | 3.2 | * |
| hcp | 9.0 | 3.0 | 15.2 | 3.1 | * |
| sig2eps | 0.0 | 0.0 | 0.0 | 0.0 | |
| sig2eta | 50,130.3 | 42,202.6 | 58,597.8 | 4,198.9 | |
| phi | 3.1 | 1.2 | 6.3 | 1.3 |
{INLA}
\(\sigma^2_{\epsilon}\) random effect error
\(\sigma^2_{\eta}\) spatial error
\(\phi\) decay parameter
| Params | mean | 2.5% | 97.5% | sd | Significant |
|---|---|---|---|---|---|
| 35_64 | −219.4 | −278.0 | −162.5 | 29.2 | * |
| 65_and_older | 219.8 | 165.9 | 275.9 | 27.8 | * |
| men | 45.0 | −11.6 | 106.6 | 29.4 | |
| women | −44.9 | −102.5 | 11.3 | 28.7 | |
| ozone | 19.4 | 17.1 | 21.9 | 1.2 | * |
| hcp | 8.0 | −6.7 | 22.7 | 7.6 | |
| sigma2eps | 348,330.8 | 335,768.1 | 362,175.8 | 6,584.7 | |
| sig2eta | 0.0 | 0.0 | 0.0 | 0.0 | |
| phi | 0.0 | 0.0 | 0.0 | 0.0 |
| model | rmse | mae | cvg | Comp_Time | Sig |
|---|---|---|---|---|---|
| spTimer | 164.1 | 137.3 | 100.0 | 3,000 | men, women, housing_change_percent |
| inla | 307.3 | 276.3 | 11.4 | 719 | housing_change_percent |
STA 5377: Spatial Statistics